AN OLG MACROECONOMIC MODEL IN A NON-NEUTRALITY MONEY CONTEXT: COMPLEX

DYNAMICS FERNANDO BIGNAMI and ANNA AGLIARI

Catholic University of Piacenza

DOMENICO DELLI GATTI and

TIZIANA ASSENZA

Catholic University of Milano

FRAMEWORK OF THE MODEL• A large empirical literature has shown that the inflation rate

could affect real activity leading to an increase in saving output or the capital stock (i.e. Loayza et al. 2000, Kahn et al. 2001). In the last decade the long run real effects of inflation have been detected even in models with financial market imperfections (i.e. Body and Smith, 1998, Cordoba and Ripoll, 2004, Ragot 2006.)

• The present paper can be classified in this framework. We present a new chanel of non-neutrality of monetary policy in the presence of financial frictions. When agents face a borrowing constraint a redistribution of real assets can occur due to the interaction between net worth and inflation. A change in the growth rate of money supply can affect real output through the impact of inflation on borrowers’ net worth.

• This model is an overlapping generation version of a Kyotaki and Moore (1997) economy, with money and bequest.

• In our model the novel feature is the role of the money as a store of value and of the bequest as a source of founds to be invested in landholding.

• In this setting we explore the properties of

the two-dimensional model that represents the

law of evolution of the economy.

OLG-KM MODEL

In each period there are four classes of agents:

YOUNG FARMER (YF)OLD FARMER (OF)YOUNG GATHERER (YG)OLD GATHERER (OG)

And two types of goods:Output (y)Non-reproducible asset (land, K) whose total supply is fixed

The production functions of the YF and of the YG are:

1

1

, 0

, . is increasing,

strictly concave and satisfies the Inada conditions.

F Ft t

G Gt t

y K

y G K G

A bequest motive is rooted in the intergenerational altruism.The generical utility function si:

, 1 1 ,

, 1

1

,,

, , ,

where is consumption of the agent of type i and generation t in t+1 (the old agent);

is bequest left by the same agent to his/her child;

: are

i i i it t t t t

it t

it

it ti

t tt

U U c a m i F G

c

a

Mm

P

real money balances of the agent of type i and generation t in t (the young agent).

FARMER’S OPTIMIZATION PROBLEM

1

,,

The YF is endowed at birth with bequest . He employs the bequest and the credit to invest in land

and hold money balance

flow-of found ( )

s : .

The constra of the YFi t n

F F Ft t t t t

Ft tF

t tt

a b

F

K

Mm

P

F

q K

1 ,

1

1

financial

(in real term

ially coinstr

s) is:

where: : is the real price of land. The YF is :

1 where : is the real gross interest rate and 1

e

1

ain d

F F F Ft t t t t t t

Ft tt t t

t

tt

t

q K K m b a

Q qq b K

P R

iR

11

, 1 1 , 1FF constra

: is the

int of th

gross rate of infl

e old farmer (OF)

ation. R is given and costant.

The is :

t

t

F F F Ft t t t t t t

P

P

c a Rb K m

The farmer are three constraints:• flow-of-found (FF) constraint when young; • FF constraint when old; • financing constraint.

• The farmer maximizes own utility function subject to three constraints:

, 1 1 ,max ln 1 ln

. FF constraint of the YF

FF constraint of the OF

Financing constraint

0< <1, 0

F F F F Ft t t t t

F

U c a m

s t

Money has two different and contrasting effects on the net worth:• Given the bequest, the higher is money of the young, the lower net worth and landholding.• The higher is money of the old, the higher resources available tohim and the higher the bequest the hold leaves to the young

GATHERER’S OPTIMIZATION PROBLEM

• Being unconstrained from the financial point of view, the gatherer maximizes own utility function subject to the FF constraint of the YG and of the OG.

, 1

, 1 1 , 1

, 1 1 11 1

FF constraint of the YG:

FF constraint of the OG:

1where , 0< <1, :

1

G G G Gt t t t t t t

G G G G Gt t t t t t t t

G G tt t t t t

t t

m b q K K a

c a CMF G K Rb m

PCMF m

P

• Then, the optimization problem is

, 1 1 ,max ln 1 ln

s.t FF constraint of the YG

FF constraint of the OG

G G G G Gt t t t tU c a m

' with Ft F

t t

g Kq g G K K

R

From F.O.C. we obtain the asset price equation

RESOURCE CONSTRAINTS AND MONEY FLOWS

• Since the total amount of land is fixed, an increase of landholding for the farmer can occur only if there is a corresponding decrease of landholding for the gatherer (aggregate resource constraint).

The sum of aggregate output and real money balances of the old agents is equal to the sum of aggregate consumption of the old agents and real money balances of the young agents.Moreover: the total amount of real money balances of the young agents is equal to the total amount of real money balances of the old agents.

• In order to describe the way in which money flows in the economy, let’s assume that the OF consumes less than the output he has produced, while the OG consumes more than the output he has produced. The OF sells units of output “savings” to the OG in order to let him consume in excess of his output. The OG pays this output by means of money. After the transaction, the OF use this money to remburse debit to the OG and leave the bequest to the YF. The YF receives this bequest from OF and credit from YG and employs these resources to invest

and holds money balances. 1 1

F Ft t tq K K

• From these considerations, we obtain an equation that represent a sort of quantity theory of money in this model.

,

,

,

1 1

1

where :

(i.e. the ratio of money of the gatherer to money of farmer of the same generation is constant)

F F Gt t t t

Gt t

Ft t

m y Y

m

m

The dynamic of the economy is described by:•The law of motion of the farmer’s land;•The asset price equation;•The quantity theory of money.Since the dimensionality of the system can be reduced, we obtain the following map T

THE MAP T

2 2

1 11

2 2

1

4

:

1

4

1+ 1 11-where A=A'

1+ 1+ 1

1+ 1and '

1+ 1

tt

t t tt t

t

k kR q

TAB q k A k k

q R qk

R q

B

A

'2 2

'

2 2

1 1

1

4

1

4

where: ; '; ; 't t t

x xR y

AB y x A x xy R y

xR y

k x k x q y q y

FIXED POINTS

2

2

0 then there exists an unique fixed point

2 1 1 4, ;

41 1 4

0 then no fixed point exists

If B

x B xE x y

BRxB x

If B

COEXISTENCE

q

k

E*1C

q

k

E

INVARIANT CLOSED CURVE

q

k

E

*1C

q

k

E

*1C

FLIP BIFURCATION SEQUENCE

q

k

E

q

k

E

STRANGE ATTRACTORS

q

k

E

q

k

E

MAP WITH DENOMINATOR

• The map of this model is a plane map with denominator. If the denominator vanish, then the map is not defined in the whole plane and some particular behaviors can be related to this fact. In particular if one of the components of the map (or of its inverse) assume the form 0/0 in a point of plane, then some particular dynamic properties of the map can be evidenced, related to the presence of this points [see i.e. Bischi, Gardini and Mira 1999].

DIFFERENT KINDS OF CONTACT BIFURCATIONS

• For this map in which the denominator vanish (or one of its inverse) can be occur different kinds of contact bifurcations. This bifurcations are explained by contacts between arc of phase curve and some singularity of the map as:

SET OF NONDEFINITION;

PREFOCAL CURVE (FOCAL POINT).

These bifurcations cause the creation of particular structures of the basin boundaries, denoted as lobes and crescents

• The set of nondefinition coincides with the locus of points in which at least one denominator vanishes.

For the map T, we obtain:

• The focal points are defined which are simple roots of the algebric system:

2 1, : 0

2s x y y y

R x

, 0

, 0

where and are numerator and denominator respectively of the map T

N x y

D x y

N D

Focal points

enlargement

q

k

E

q

k

E

1Q

• The prefocal curve is a set of points for which at least one inverse exist which focalizes the whole set into a single point, called focal point [Mira, 1996]. For the map T the prefocal curve is located on the y axis.

• The bifurcations due to tangential contacts between arcs of phase curve and a prefocal curve or a set nondefinition are denoted as bifurcations of first class

• The bifurcations due to the merging of focal points or to the merging of focal points and fixed points, or to contacts between prefocal curves and critical curves are denoted as bifurcations of second calss

Disappearance of the closed invariant curve

q

k

q

k

E

*1C

*1C

Map T5

REFERENCES• G.I. Bischi, L. Gardini and C. Mira, 1999. Plane Maps with

Denomiantor. I. Some Geometric Properties, International Journal of Bifurcation and Chaos, 9, 119-153.

• J. Body, B. Smith, 1998. Capital Market Imperfections in a Monetary Growth Model, Economy Theory, 11, 241-273.

• M. Cordoba, J.C. Ripoll, 2004. Collateral Constraints in a Monetary Economy, Journal of European Economic Association, 2, n° 6, December

• N. Kiyotaky, J. Moore, 1997. Credit Cycle, Journal of Political Economy, 105, 211-248.

• C. Mira, 1996. Some Properties of Two Dimensional Maps not Defined in the Whole Plane, Proc. ECIT 96 Urbino, in Grazer Mathematische Berichte.

• X. Ragot, 2006. A Theory of Law Inflation with Credit Constraints, mimeo.